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Vortex lattice structure dependent on pairing symmetry in Rashba superconductorsVortex lattice structures in Rashba noncentrosymmetric superconductors in magnetic fields parallel to the basal plane ${\bf H} \perp c$ are examined based on the BCS-like Hamiltonian and the resulting Ginzburg-Landau functional. Due to the momentum dependent anisotropy of the Zeeman effect induced by the broken inversion symmetry, the vortex lattice in higher fields generally shows some unidirectional modulation of Fulde-Ferrell-Larkin-Ovchinnikov FFLO type orienting in the plane perpendicular to ${\bf H}$. However, the direction of the modulation and the lattice structure depend significantly on the underlying pairing symmetry: When the mixing between spin singlet and triplet pairing components is negligible, the resulting modulated structure tends to have reflection symmetry, while the vortex lattice in systems with a significant singlet-triplet mixing has no reflection symmetry in most cases. The latter result implying the presence in {\it real} materials of two degenerate orientations of the lattice structure separated by domain walls may be relevant to the extremely low magnetic decay rate observed in CePt$ 3$Si.